An Introduction to Filters—What’s So Special About This Partially Ordered Set

An Introduction to Filters—What’s So Special About This Partially Ordered Set?

AGT isn’t new math that’s intended to replace existing mathematical theory, but rather a novel way to express mathematical concepts which were previously inexpressible. I indulge these ideas because I believe, as is the modus operandi for all academics, that any worthwhile, logically consistent idea must be pursued until it is falsified or verified for good.

My AGT and language are admittedly novel, but allow me and will allow mathematicians to speak of mathematical ideas that couldn’t be spoken of before. These ideas are based on the concept of filters that, incidentally, also allows me to develop AGT.

 AGT is a purely intellectual pursuit because it isn’t tainted with the obsession that institutional knowledge must exist as a continuum. That idea is limited, short-sighted and not true to the spirit of mathematical theorization or the pursuit of knowledge.

Filters—A Basic Introduction

In a rudimentary sense, filters are relations we apply to existing sets to express otherwise inexpressible statements. For example, expressing the infinitely small neighborhood around 0 is impossible, but it might be possible if we work with filters. Filters effectively allow us to refer to infinitesimally small or infinitely large sets and conduct mathematical analysis to develop valuable insights.

Filtrators

I am not concerned filters through, which are a fairly well-known idea in mathematics— I am more interested in Filtrators, which is my sole contribution to existing topological theory. Both filters and Filtrators are one of the core components of Algebraic General Topology.

 I define a filtrator as:

A pair (B, Z) of a poset B and its subset Z ⊆ B

Where

B is the base of the filtrator

Z the core of the filtrator

Some other definitions of note are:

  1. I will call a lattice filtrator a pair (B, Z) of a lattice B and its subset Z ⊆ B.
  2. I will call a complete lattice filtrator a pair (B, Z) of a complete lattice A and its subset Z ⊆ B.
  3. I will call a central filtrator a filtrator (B, Z (B)) where Z(B) is the center of a bounded lattice A.
  4. I will call an element of a filtrator an element of its base.

To quote Fermat, “I have discovered a truly remarkable proof of this theorem which this margin is too small to contain”. I too, like Fermat, find myself constrained for want of space to discuss my entire theory but these are some of the more basic definitions I introduce. Defined this way, it’s possible to express sets like the infinitesimally small set on the real line around 0 and many other similar ideas, which may have great value for mathematical research.

If you’re interested and willing to consider these ideas as a purely intellectual pursuit, you should look up on my Algebraic General Topology book Volume 1. It goes at length into Algebraic General Topology, the basic concepts involved and a collection of all the definitions used to construct the AGT. I am certain, you will find it a joyful exercise.

Learn more about general topology monograph here.

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